The following function gives the temperature (in degrees Celsius) at the beach in Miami, Florida, $t$ hours after midnight on a certain day: $M(t)=-6\cdot\text{sin}\left(\dfrac{\pi}{12}t\right)+18$ What is the instantaneous rate of change of the temperature at $9\text{ a.m.}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $1.11$ degrees Celsius per hour (Choice B) B $1.11$ degrees Celsius (Choice C) C $13.76$ degrees Celsius per hour (Choice D) D $13.76$ degrees Celsius
Answer: Understanding the problem The derivative, $M'(t)$, gives the instantaneous rate of change of $M(t)$. Therefore, the instantaneous rate of change of the temperature at $9\text{ a.m.}$ is $M'(9)$. Let's find $M'(t)$ and evaluate it at $t=9$. Finding $M'(t)$ $M'(t)=-\dfrac{\pi}{2}\cdot\text{cos}\left(\dfrac{\pi}{12}t\right)$ Finding $M'(9)$ $\begin{aligned} M'({9})&=-\dfrac{\pi}{2}\cdot\text{cos}\left(\dfrac{\pi}{12}({9})\right) \\\\ &=-\dfrac{\pi}{2}\cdot\text{cos}\left(\dfrac{3\pi}{4}\right) \\\\ &\approx 1.11 \end{aligned}$ Interpreting units $M(t)$ is the temperature in ${\text{degrees Celsius}}$ at $t$ ${\text{hours}}$. Therefore, we measure its rate of change in ${\text{degrees Celsius}}$ per ${\text{hour}}$. In conclusion, the instantaneous rate of change of the temperature at $9\text{ a.m.}$ is $1.11$ degrees Celsius per hour. The rate of change is positive because the temperature is increasing.